Tomography is used in medicine, archaeology, biology, geophysics, materials science, electron microscopy, security scanning, industrial nondestructive testing, astronomy and others. Tomography is imaging by sections or sectioning to convey internal structures of a solid object, for example the human body or the earth. Slices of the object are viewed without physically cutting the object. A device used in tomography is called a tomograph. A tomograph generates a tomogram, or image.
The image, or tomogram, can be achieved by tomography applications such as atom probe tomography (APT), computed tomography (CT), confocal laser scanning microscopy (LSCM), cryo-electron tomography (Cryo-ET), electrical capacitance tomography (ECT), electrical resistance tomography (ERT), electrical impedance tomography (EIT), functional magnetic resonance imaging (fMRI), magnetic induction tomography (MIT), magnetic resonance imaging (MRI), formerly known as magnetic resonance tomography (MRT), neutron tomography, optical coherence tomography (OCT), optical projection tomography (OPT), process tomography (PT), positron emission tomography (PET), quantum tomography, single photon emission computed tomography (SPECT), seismic tomography, and X-ray tomography.
In each tomography application, a source emits beams onto or from the object, which are then collected by a detector system. Beams include X-rays, gamma rays, positron electron annihilation reaction, radio frequency, nuclear magnetic resonance, ultrasound, electrons, ions, electron beams, radio pharmaceuticals, light, microwaves, magnetic field, to name a few. Beams are emitted with a scanning geometry such as parallel beams and circular trajectory beams, divergent beams (fan or cone) and circular trajectory beams, cone beams and helical trajectory beams, for example.
For example, in a transmission computed tomography system, a source projects a beam with scanning geometry that passes through the object, and transmits upon an array of detectors. The intensity of the transmitted radiation is dependent upon the attenuation of the object. As another example, in some emission computed tomography systems, the distribution of radio-isotopes within an object is transmitted upon an array of detectors. The detectors produce projection data, which is a separate electrical signal resultant from scanning geometry.
The projection data is then sent to a computer for processing to generate a two-dimensional or three-dimensional image, or reconstructed image. In most cases the computer processes the data based on the mathematical procedure called tomographic reconstruction. Tomographic reconstruction includes algorithms that manipulate the projection data to produce the image.
Many different reconstruction algorithms exist. Most algorithms fall into one of two categories: filtered back projection (FBP) and iterative reconstruction (IR). Usually, these algorithms provide inexact results: they represent a compromise between accuracy and computational time required. FBP demands fewer computational resources, while IR generally produces fewer artifacts. Artifacts are errors resultant from the reconstruction. FBP includes analytical methods, such as those that involve Fourier transform. IR includes algebraic methods and probabilistic methods.
Analytical methods are the fastest with respect to computational time required, whereas probabilistic methods are slower but provide more accurate results. Algebraic methods provide a relatively fast computational time along with accurate results.
Ideally, projection data should be collected as finely sampled as possible and covering a full angular range. The full angular range can be, for example, 180 degrees or 360 degrees. Projection data is a collection of one or more projections. A projection is commonly referred to the data collected in one angle. In practice it is very often that projection data are sparsely sampled or limited to a certain angular range. Limited data tomography refers to either (1) that the projection data is acquired within a limited (i.e., less than the full) angular range (“limited angle data”) or (2) that a few number of projections are acquired (“sparse projection data”).
Algorithms used in limited data tomography results in errors, or artifacts within the reconstructed image. Limited angle data tomography typically causes “fan-shape”or “butterfly” artifacts within the reconstructed image. Sparse projection data tomography typically causes “star-pattern” artifacts within the reconstructed image.
In dealing with the limited data tomography, many algorithms have been proposed such as Fourier methods, sinogram methods, regularization methods, Bayesian method, wavelet techniques, etc. Most of these algorithms are vulnerable to artifacts, unless a “reference image” is used, which is not always available.
There is a desire to provide an algorithm that improves quality and accuracy within a reconstructed image by reducing or eliminating artifacts. There is also a desire to provide an algorithm that can be applied to many tomography applications.